The range of the integration map of a vector measure
Let be \(X\) a locally convex Hausdorff space, \(m\) a countably additive \(X\)-valued measure, \(X [m]\) the sequential closure of the linear span of the range \(m (\Sigma)\), \(L^1 (m)\) the space of the \(\mathbb{C}\)-valued \(m\)- integrable functions and \(I_m : L^1 (m) \to X\) the integration map. Several criteria are stated which ensure that \(I_m (L^1(m))\) is a sequentially closed subspace of \(X\) (in which case \(I_m (L^1(m)) = X[m])\). Also it is proved that if \(X\) is Fréchet, \(I_m\) is weakly compact and \(I_m (L^1(m))\) is infinite dimensional then \(I_m (L^1(m))\) is not closed.
scientific article; zbMATH DE number 769538
scientific article